Residue currents of holomorphic morphisms

Given a generically surjective holomorphic vector bundle morphism $f\colon E\to Q$,$E$and$Q$Hermitianbundles,weconstructacurrent$R^f$ with values in $\text{Hom}(Q,H)$, where $H$ is a certain derived bundle, and with support on the set $Z$ where $f$ is not surjective. The main property is that if $\varphi$ is a holomorphic section of $Q$, and $R^f\varphi =0$, then locally $f\psi =\varphi$ has a holomorphic solution $\psi$. In the generic case also the converse holds. This gives a generalization of the corresponding theorem for a complete intersection, due to Dickenstein-Sessa and Passare. We also present results for polynomial mappings, related to M Noether’s theorem and the effective Nullstellensatz. The construction of the current is based on a generalization of the Koszul complex. By means of this complex one can also obtain new global estimates of solutions to $f\psi =\varphi$, and as an example we give new results related to the $H^p$-corona problem.